1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
© Gary Parker November, 2004
CHAPTER 1:
FELIX M. EXNER AND THE ORIGINS OF MORPHODYNAMICS
Felix Maria Exner was an Austrian
researcher who was active in the early
part of the 20th Century. His main area
of interest was meteorology. At some
point he became interested in the
formation of dunes in rivers (Exner,
1920, 1925; see also Leliavsky, 1966).
In the course of his research on the
subject, he derived and employed one
version of the various statements of
conservation of bed sediment that are
now referred to as “Exner equations.”
In addition, he made an important early
contribution to 1D nonlinear wave
dynamics.
Dunes in the Mississippi River, New
Orleans, USA
Image from LUMCON web page:
1
http://weather.lumcon.edu/weatherdata/audubon/map.html
1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
© Gary Parker November, 2004
DUNE ASYMMETRY
Dunes are rhythmic bedforms that are
often seen on the beds of rivers. The
upstream (stoss) side of a dune has a
gentle slope, whereas the downstream
(lee) side has a steep slope. This
characteristic asymmetry allows the
determination of flow direction.
Flow
In the image on the right, the flow
direction is from top to bottom. Dunes
migrate in the same direction as the
flow, i.e. downstream.
flow
Magnified view of previous image
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1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
© Gary Parker November, 2004
DUNES IN THE RHINE DELTA, THE NETHERLANDS
Image courtesy A. Wilbers and A. Blom
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1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
© Gary Parker November, 2004
DUNES IN THE LABORATORY
Flow
Flow
Dunes in a channel at St. Anthony
Falls Laboratory, University of
Minnesota, USA
Dunes in a channel at Tsukuba
University, Japan.
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Image courtesy H. Ikeda.
1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
© Gary Parker November, 2004
EXNER’S QUESTION:
WHY ARE DUNES ASYMMETRIC?
The parameters:
x = streamwise distance [L]
t = time [T]
= bed elevation [L]
qt = volume total sediment transport
rate per unit stream width [L2/T]
p = bed porosity [1]
g = acceleration of gravity [L/T2]
H = flow depth [L]
U = depth-averaged flow velocity [L/T]
Cf = bed friction coefficient [1]
U
H

x
The notation in brackets denotes dimensions: M denotes mass, L denotes length
and T denotes time. The bed friction coefficient is defined such that b = CfU2,
where b denotes bed shear stress [M/L/T2] and  denotes water density [M/L3].
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1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
© Gary Parker November, 2004
THE EQUATIONS
St. Venant shallow water equations:
H UH

0
t
x
UH U2H
1
H


  gH
 gH
 Cf U2
t
x
2
x
x
These are statements of conservation
of water mass and momentum in a 1D
river (constant width).
Exner’s equation of conservation of bed
sediment:
(1   p )
q

 t
t
x
Relation between sediment transport rate
and flow hydraulics:
qt  qt (U)
Exner’s seminal contribution: if more
sediment enters a reach than leaves, the
bed elevation in the reach increases.
The phenomenon of sediment transport
was poorly known in Exner’s time. Exner
guessed that a higher velocity caused a
6
higher sediment transport rate.
1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
© Gary Parker November, 2004
THE RESULTS
The details of the analysis are not considered here. The essential result is a
nonlinear equation for the evolution of bed elevation  of the general form


 c()
 (other terms )
t
x
where c() denotes a positive wave speed of the bed that is an increasing function of
bed elevation. The result is that any symmetric bedform sharpens in time to the
asymmetry characteristic of dunes.
Image from Leliavsky
(1966)
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1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
© Gary Parker November, 2004
FELIX EXNER AND MORPHODYNAMICS
The field of morphodynamics consists of the class of problems for which the flow
over a bed interacts strongly with the shape of the bed, both of which evolve in time.
That is, the flow and the bed talk to each other. The flow field over the bed
determines a pattern of variation of sediment transport rate. This variation changes
the bed by erosion or deposition of sediment. The changed bed now induces a
changed flow field.
H UH

0
t
x
UH U2H
1
H


  gH
 gH
 Cf U2
t
x
2
x
x
qt

(1   p )

t
x
qt  qt (U)
Felix Exner was the first researcher to state a morphodynamic problem in quantitative
terms. The term “morphodynamics” itself evolved many decades afterward. This 8
notwithstanding, Exner deserves recognition as the founder of morphodynamics.
1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
© Gary Parker November, 2004
DERIVATION OF EXNER’S EQUATION OF SEDIMENT CONSERVATON
Recall that qt denotes the volume sediment transport rate per unit width and p
denotes bed porosity (fraction of bed volume that is pores rather than sediment).
The mass sediment transport rate per unit width is then sqt, where s is the
material density of sediment. Mass conservation within the control volume with a
unit width requires that:
/t (sediment mass in bed) =
mass sediment inflow rate –
mass sediment outflow rate
water
qqb t
or



s (1   p )x  1  s qt x  qt
t
x  x
qqbt
 1

or
qt

(1   p )
t
x
bed sediment + pores
1
x
9
x
x +x
1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
© Gary Parker November, 2004
1D SEDIMENT TRANSPORT MORPHODYNAMICS
While Exner’s 1D model successfully explains a) the tendency of dunes to migrate
downstream and b) their tendency to become asymmetric, it does not explain the
origin of dunes. Such an explanation requires a 2D analysis, and as such is not
considered further here. The interested reader can refer to e.g. Engelund (1970),
Smith (1970) and Fredsoe (1974).
Dunes are one of many examples of fascinating
morphodynamic problems requiring 2D or 3D
approaches. For example, the pattern of meandering
and sediment sorting in the river on the right cannot be
explained with a 1D approach.
So why is this book limited to 1D morphodynamics?
1. There are many interesting 1D problems.
2. A surprising number of 2D problems can
be reduced and explained using a 1D
formulation.
3. 1D morphodynamics is the gateway to 2D
and 3D morphodynamics.
Ooi River, Japan.
10
Image courtesy H. Ikeda
1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
© Gary Parker November, 2004
RIVER STRAIGHTENING: A PROBLEM IN 1D MORPHODYNAMICS
Consider a meandering river.
flow
A
In 1D approaches, the relevant streamwise
coordinate is measured as an arc length down
the channel centerline. The long profile of the
river is as illustrated below. The fluctuations
are induced by the meanders themselves.

B
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x (arc length coordinate)
1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
© Gary Parker November, 2004
RIVER STRAIGHTENING (contd.)
Now suppose the bends between points A and B are cut off due to river straightening.
B
A
flow
ambient slope
A
steepened slope

The elevation drop between A and B
is the same as before straightening
(at least initially), but the distance
between A and B is much shorter. As
a result bed slope is elevated
between A and B compared to the
reaches upstream and downstream.
ambient slope
12
x (arc length coordinate)
1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
© Gary Parker November, 2004
RIVER STRAIGHTENING (contd.)
Now let’s exaggerate the change in bed slope for the sake of illustration.
U
A
The bed slope is higher, the flow
swifter and the sediment transport
rate higher over the steepened reach.

B
D
x
Exner equation of sediment conservation over a reach with length x (set porosity
p = 0 for illustration).
q

 t
t
x

 t  t   t 

1
q x  q x  x
x

Bed elevation of the reach increases in time if more sediment enters than
leaves.
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1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
© Gary Parker November, 2004
RIVER STRAIGHTENING (contd.)
U
A
qt


t
x

 t  t

1
 t 
qt x  qt
x
x  x
t
The sediment transport rate
is higher over the
steepened reach.

B
D
qt
x
The reach UA must degrade (bed elevation must decrease in time), as there is
more sediment output than input.
The reach between B and D must aggrade (bed elevation must increase in time),
as there is more sediment input than output.
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1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
© Gary Parker November, 2004
RIVER STRAIGHTENING (contd.)
Initial long profile after straightening.
U
A
Evolving long profile.

B
D
Final equilibrium long
profile: qt/x = 0 thus
/t = 0
x
The upstream part of the reach must degrade and the downstream part must
aggrade until a new equilibrium profile is established.
The above 1D picture is a simplified version of reality: too much aggradation
downstream will cause the river to avulse (jump channel). Nevertheless, it
captures the essentials of the problem.
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1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
© Gary Parker November, 2004
AN EXAMPLE: THE EAST PRAIRIE RIVER, ALBERTA, CANADA
River straightening was popular in the 1960’s and 1970’s. The goal was flood
protection: the steeper slope of the straightened reach allowed higher flow
velocities, and thus conveyance of the same discharge at a lower depth. This
short-term benefit was often outweighed by the long-term pattern of aggradation
and degradation induced by straightening. The East Prairie River provides such
an example. See Parker and Andres (1976) for more details.
Degrading upstream
reach
Straightened central
reach
Aggrading downstream
reach
East Prairie River, Alberta, Canada
Images courtesy D. Andres
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1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
© Gary Parker November, 2004
THE STRAIGHTENED WEST AND EAST PRAIRIE RIVERS, ALBERTA, CANADA
West Prairie River
East Prairie River
straightened
Straightened
central
meandering
reach
Image from NASA
https://zulu.ssc.nasa.gov/mrsid/mrsid.pl
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1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
© Gary Parker November, 2004
REFERENCES FOR CHAPTER 1
Engelund, F., 1970, Instability of erodible beds, J. Fluid Mech., 42(2), 225-244.
Exner, F. M., 1920, Zur Physik der Dunen, Sitzber. Akad. Wiss Wien, Part IIa, Bd. 129 (in
German).
Exner, F. M., 1925, Uber die Wechselwirkung zwischen Wasser und Geschiebe in Flussen,
Sitzber. Akad. Wiss Wien, Part IIa, Bd. 134 (in German).
Fredsoe, J., 1974, On the development of dunes in erodible channels, J. Fluid Mech., 64(1), 116.
Leliavsky, S., 1966, An Introduction to Fluvial Hydraulics, Dover, New York, 257 p.
Parker, G. and Andres, D., 1976, Detrimental effects of river channelization, Proceedings, ASCE
Rivers '76 Conference, 1248-1266.
Smith, J. D., 1970, Stability of a sand wave subjected to shear flow of low Froude number, J.
Geophys. Res., 75(30), 5928-5940.
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